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arXiv · 1204.4501

Discrete Fourier Analysis and Chebyshev Polynomials with $G_2$ Group

Huiyuan Li·Jiachang Sun·Yuan Xu
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Abstract

The discrete Fourier analysis on the $30^{\degree}$-$60^{\degree}$-$90^{\degree}$ triangle is deduced from the corresponding results on the regular hexagon by considering functions invariant under the group $G_2$, which leads to the definition of four families generalized Chebyshev polynomials. The study of these polynomials leads to a Sturm-Liouville eigenvalue problem that contains two parameters, whose solutions are analogues of the Jacobi polynomials. Under a concept of $m$-degree and by introducing a new ordering among monomials, these polynomials are shown to share properties of the ordinary orthogonal polynomials. In particular, their common zeros generate cubature rules of Gauss type.

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Huiyuan Li, Jiachang Sun, Yuan Xu. 2012-04-19. Discrete Fourier Analysis and Chebyshev Polynomials with $G_2$ Group. https://doi.org/10.3842/sigma.2012.067

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